Let P=[p_ij] and Q=[q_ij] be two square matrices of order 3 such that q_ij=2^(i+j-1)p_ij a…

Question

Let $P=[{p}_{ij}]$ and $Q=[{q}_{ij}]$ be two square matrices of order 3 such that ${q}_{ij}={2}^{(i+j-1)}{p}_{ij}$ and $\det (Q)={2}^{10}$. Then the value of $\det (adj(adjP))$ is:

Accepted Answer

Given two $3	imes 3$ matrices $P=[{p}_{ij}],Q=[{q}_{ij}]$ are given

${q}_{ij}={2}^{(i+j-1)}{p}_{ij},\det (Q)={2}^{10}.$

To calculate $\det (adj(adjP))$

We know $\det (adj(adjP))=(\det (P){)}^{(n-1{)}^{2}}$

where $n$ is order of matrix which is 3

so, $\det (adj(adjP))=(\det (P){)}^{4}-(1)$

Let P matrix

$P=[[\begin{vmatrix} {p}_{11} \quad  {p}_{12} \quad  {p}_{13} \ {p}_{21} \quad  {p}_{22} \quad  {p}_{23} \ {p}_{31} \quad  {p}_{32} \quad  {p}_{33} \end{gathered}]]|$

and $Q=[[|\begin{gathered} {q}_{11} \quad  {q}_{12} \quad  {q}_{13} \ {q}_{21} \quad  {q}_{22} \quad  {q}_{23} \ {q}_{31} \quad  {q}_{32} \quad  {q}_{33} \end{gathered}]]|$

use ${q}_{ij}={2}^{i+j-1}{p}_{ij}$ to find Q matrix

${q}_{11}={2}^{1+1-1}{p}_{11}=2{p}_{11},{q}_{23}={2}^{2+3-1}{p}_{23}={2}^{4}{p}_{23}$

$Q=[[|\begin{gathered} 2{p}_{11} \quad  {2}^{2}{p}_{12} \quad  {2}^{3}{p}_{13} \ {2}^{2}{p}_{21} \quad  {2}^{3}{p}_{22} \quad  {2}^{4}{p}_{23} \ {2}^{3}{p}_{31} \quad  {2}^{4}{p}_{32} \quad  {2}^{5}{p}_{33} \end{gathered}]]|$

$\begin{gathered} Q=[[|\begin{array}{ccc}2{p}_{11} \quad  {2}^{2}{p}_{12} \quad  {2}^{3}{p}_{13} \ {2}^{2}{p}_{21} \quad  {2}^{3}{p}_{22} \quad  {2}^{4}{p}_{23} \ {2}^{3}{p}_{31} \quad  {2}^{4}{p}_{32} \quad  {2}^{5}{p}_{33}\end{array}]]| \ \det (Q)=|\begin{array}{ccc}2{p}_{11} \quad  {2}^{2}{p}_{12} \quad  {2}^{3}{p}_{13} \ {2}^{2}{p}_{21} \quad  {2}^{3}{p}_{22} \quad  {2}^{4}{p}_{23} \ {2}^{3}{p}_{31} \quad  {2}^{4}{p}_{32} \quad  {2}^{5}{p}_{33}\end{array}|={2}^{10} \end{gathered}$

taking common 2 from ${R}_{1},{2}^{2}$ from ${R}_{2}\&{2}^{3}$ from ${R}_{3}$

$\det (Q)=2\cdot {2}^{2}\cdot {2}^{3}|\begin{gathered} {p}_{11} \quad  2{p}_{12} \quad  {2}^{2}{p}_{13} \ {p}_{21} \quad  2{p}_{22} \quad  {2}^{2}{p}_{23} \ {p}_{31} \quad  2{p}_{32} \quad  {2}^{2}{p}_{33} \end{vmatrix}={2}^{10}$

taking common 2 from ${C}_{2}$ and ${2}^{2}$ from ${C}_{3}$

$2\cdot {2}^{2}\cdot {2}^{3}\cdot 2\cdot {2}^{2}\begin{vmatrix} {p}_{11} \quad  {p}_{12} \quad  {p}_{13} \ {p}_{21} \quad  {p}_{22} \quad  {p}_{23} \ {p}_{31} \quad  {p}_{32} \quad  {p}_{33} \end{vmatrix}={2}^{10}$

$\begin{gathered} \Rightarrow {2}^{9}\cdot \det (P)={2}^{10} \ \Rightarrow \det (P)=2 \ (1)\det (adj(adjP))=(\det (P){)}^{4}={2}^{4}=16 \end{gathered}$

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